How do cortical populations represent sensory input and support perceptual decision making? It has long been known that the responses of individual neurons to the repeated presentation of a stimulus are highly variable. Nonetheless, the pattern of activity across a population encodes enough information to support precise perceptual decisions. This implies that hidden in the distribution of population responses there are invariant features, yet to be identified, which robustly encode the sensory stimulus from one trial to the next. Here we propose to mathematically model the conditional distribution of the population response given a stimulus and to uncover the invariant features that support the reliable discrimination of sensory stimuli. Surprisingly, preliminary data reveal that the distribution of population responses to a fixed stimulus is star- shaped ? in any one trial, the population vector can point in one among a finite set of directions. The directions are highly invariant across trials, while the amplitude of the responses is variable. Based on these observations we hypothesize that cortical coding is a one-to-many correspondence. This idea represents a major departure from the prevailing view of cortical coding as a one-to-one map between a stimulus and a population direction. We propose to study star-shaped distributions and their role in the encoding of sensory information with the following three Aims: (a) measure population responses to the repeated presentation of a stimulus and test the hypothesis the structure is well-described by star-shaped distributions, (b) test a mathematical model linking the direction of population responses evoked in indiviudal trials to behavioral choice in a discrimination task, (c) establish if star-shaped distributions are generated in the cortex or inherited from thalamic input. The proposed studies are significant because they challenge the dominant view that cortical coding implements a one-to-one map. We introduce an innovative framework to study and understand the structure of cortical variability that generalizes prior approaches, yielding predictive models that link population activity to behavior. Altogether, the proposed studies will significantly advance our understanding of cortical coding and function.